论文标题
3属的非高温曲线家族的2-核子组的平均大小
The average size of the 2-Selmer group of a family of non-hyperelliptic curves of genus 3
论文作者
论文摘要
我们表明，非Hyperelliptic属的雅各布人家族的平均尺寸为$ 3 $ curves，当自然高度订购时，具有明显的理性超屈服点的平均尺寸在上面的订购时为$ 3 $。我们通过将$ 2 $ -SELMER元素解释为与稳定的$ \ Mathbb {Z}/2 \ Mathbb {Z} $相关的表示形式的积分轨道来实现这一目标。我们使用此结果表明，明显的点是该家族曲线正相位比例的唯一理性点。主要的新颖性是使用$ e_6 $的简单曲线奇异性的压实雅各布的某些属性来构建积分代表，以及与我们的曲线家族自然相关的Mumford Theta群体的表示理论解释。
We show that the average size of the $2$-Selmer group of the family of Jacobians of non-hyperelliptic genus-$3$ curves with a marked rational hyperflex point, when ordered by a natural height, is bounded above by $3$. We achieve this by interpreting $2$-Selmer elements as integral orbits of a representation associated with a stable $\mathbb{Z}/2\mathbb{Z}$-grading on the Lie algebra of type $E_6$ and using Bhargava's orbit-counting techniques. We use this result to show that the marked point is the only rational point for a positive proportion of curves in this family. The main novelties are the construction of integral representatives using certain properties of the compactified Jacobian of the simple curve singularity of type $E_6$, and a representation-theoretic interpretation of a Mumford theta group naturally associated to our family of curves.
